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<article class="li"><h3 class="heading">
<span class="type">Item</span><span class="period">.</span>
</h3>
<p><dfn class="terminology">Convolution</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\cal L}[f(t)*g(t)]=F(s) G(s),
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(f(t)*g(t)=\int_0^t f(t-\tau) g(\tau) \, \mathrm{d}\,\tau=\int_0^t f(\tau) g(t-\tau) \, \mathrm{d}\,\tau\text{.}\)</span>Example:</p>
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\begin{equation*}
{\mathcal L}^{-1}\left[ \frac{f(s)}{s} \right].
\end{equation*}
</div>
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\begin{equation*}
{\mathcal L}^{-1}[f(s)]=f(t), \ \ {\rm and \ } {\mathcal L}^{-1}\left[\frac{1}{s}\right]=1=g(t),
\end{equation*}
</div>
<p class="continuation">so</p>
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\begin{equation*}
{\mathcal L}^{-1}\left[ \frac{f(s)}{s} \right]=\int_0^t f(\theta) \, d\theta.
\end{equation*}
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